data clusteringangom.myweb.cs.uwindsor.ca/.../592-st-nsb-dataclustering.pdf5 data clustering find...

DATA CLUSTERING

Dr. Alioune Ngom

School of Computer Science

University of Windsor

[email protected]

Winter 2013

Clustering

Cluster: a collection of data objects Similar to one another within the same cluster

Dissimilar to the objects in other clusters

Cluster analysis Grouping a set of data objects into clusters

Clustering is unsupervised classification: no predefined classes

Typical applications to get insight into data

as a preprocessing step

we will use it for image segmentation

Why clustering? 3

A cluster is a group of related objects

In biological nets, a group of “related” genes/proteins

Application in PPI nets:

Protein function prediction

Protein complex identification

Are you familiar with Gene Ontology?

Clustering 4

Data clustering (Lecture 6) vs. Graph clustering

5

Data Clustering

Find relationships and patterns in the data

Get insights in underlying biology

Find groups of “similar” genes/proteins/samples

Deal with numerical values of biological data

They have many features (not just color)

6

Data Clustering

(homogeneity)

(separation)

7

There are many possible distance metrics between

objects

Theoretical properties of distance metrics, d:

d(a,b) >= 0

d(a,a) = 0

d(a,b) = 0 a=b

d(a,b) = d(b,a) – symmetry

d(a,c) <= d(a,b) + d(b,c) – triangle inequality

Data Clustering

8

Example distances:

Euclidean (L2) distance

Manhattan (L1) distance

Lm: (|x1-x2|m+|y1-y2|

m)1/m

L∞: max(|x1-x2|,|y1-y2|)

Inner product: x1x2+y1y2

Correlation coefficient

For simplicity, we will concentrate on Euclidean distance

Clustering Algorithms – Review

Distance/Similarity matrices:

• Clustering is based on distances –

distance/similarity matrix: • Represents the distance between objects

• Only need half the matrix, since it is symmetric

Clustering Algorithms – Review

9

What is Cluster Analysis?

Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups

Inter-cluster distances are maximized

Intra-cluster distances are

minimized

Notion of a Cluster can be Ambiguous

How many clusters?

Four Clusters Two Clusters

Six Clusters

Types of Clusters: Contiguity-Based

Contiguous Cluster (Nearest neighbor or Transitive) A cluster is a set of points such that a point in a cluster is closer (or

more similar) to one or more other points in the cluster than to any point not in the cluster.

8 contiguous clusters

Types of Clusters: Density-Based

Density-based A cluster is a dense region of points, which is separated by low-

density regions, from other regions of high density.

Used when the clusters are irregular or intertwined, and when noise and outliers are present.

6 density-based clusters

Euclidean Density – Cell-based

Simplest approach is to divide region into a number

of rectangular cells of equal volume and define

density as # of points the cell contains

Euclidean Density – Center-based

Euclidean density is the number of points within a

specified radius of the point

Data Structures in Clustering

Data matrix

(two modes)

Dissimilarity matrix

(one mode)

npx...nfx...n1x

...............

ipx...ifx...i1x

...............

1px...1fx...11x

0...)2,()1,(

:::

)2,3()

...ndnd

0dd(3,1

0d(2,1)

0

Interval-valued variables

Standardize data

Calculate the mean squared deviation:

where

Calculate the standardized measurement (z-score)

Using mean absolute deviation could be more robust than using

standard deviation

.)...21

1nffff

xx(xn m

)2||...2||2|(|121 fnffffff

mxmxmxns

f

fif

if s

mx z

Similarity and Dissimilarity Between Objects

Euclidean distance:

Properties

d(i,j) 0

d(i,j) = 0 iff i=j

d(i,j) = d(j,i)

d(i,j) d(i,k) + d(k,j)

Also one can use weighted distance, parametric Pearson

product moment correlation, or other disimilarity measures.

)||...|||(|),( 22

22

2

11 pp jx

ix

jx

ix

jx

ixjid

The set of 5 observations, measuring 3 variables,

can be described by its mean vector and covariance matrix.

The three variables, from left to right are

length, width, and height of a certain object, for example.

Each row vector Xrow is another observation

of the three variables (or components) for row=1, …, 5.

Covariance Matrix

The mean vector consists of the means of each variable. The covariance matrix

consists of the variances of the variables along the main diagonal and the

covariances between each pair of variables in the other matrix positions.

0.025 is the variance of the length variable,

0.0075 is the covariance between the length and the width variables,

0.00175 is the covariance between the length and the height variables,

0.007 is the variance of the width variable.

where n = 5

for this example

n

row

krowkjrowjjk

n

row

rowrow

xXxXn

s

xXxXn

XXn

S

1

1

))((1

1

)')((1

1'

1

1

Mahalanobis Distance

n

i

kikjijkj XXXXn 1

, ))((1

1

Tqpqpqpsmahalanobi )()(),( 1

For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6.

is the covariance matrix of

the input data X

Mahalanobis Distance

3.02.0

2.03.0

Covariance Matrix:

B

A

C

A: (0.5, 0.5)

B: (0, 1)

C: (1.5, 1.5)

Mahal(A,B) = 5

Mahal(A,C) = 4

Cosine Similarity

If x1 and x

2 are two document vectors, then

cos( x1, x

2 ) = (x

1 x

2) / ||x

1|| ||x

2|| ,

where indicates vector dot product and || d || is the length of vector d.

Example:

x1 = 3 2 0 5 0 0 0 2 0 0

x2 = 1 0 0 0 0 0 0 1 0 2

x1 x2= 3*1 + 2*0 + 0*0 + 5*0 + 0*0 + 0*0 + 0*0 + 2*1 + 0*0 + 0*2 = 5

||x1|| = (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5 = (42) 0.5 = 6.481

||x2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2) 0.5 = (6) 0.5 = 2.245

cos( x1, x

2 ) = .3150

Correlation

Correlation measures the linear relationship between

objects

To compute correlation, we standardize data

objects, p and q, and then take their dot product

)(/))(( pstdpmeanpp kk

)(/))(( qstdqmeanqq kk

qpqpncorrelatio ),(

Visually Evaluating Correlation

Scatter plots

showing the

similarity from

–1 to 1.

K-means Clustering

Partitional clustering approach

Each cluster is associated with a centroid (center point)

Each point is assigned to the cluster with the closest centroid

Number of clusters, K, must be specified

The basic algorithm is very simple

k-means Clustering

An algorithm for partitioning (or clustering) N data

points into K disjoint subsets Sj containing Nj data

points so as to minimize the sum-of-squares criterion

2

1

|| j

K

j Sn

n

j

xJ

where xn is a vector representing the nth data point and j is

the geometric centroid of the data points in Sj

K-means Clustering – Details

Initial centroids are often chosen randomly.

Clusters produced vary from one run to another.

The centroid is (typically) the mean of the points in the cluster.

‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc.

K-means will converge for common distance functions.

Most of the convergence happens in the first few iterations.

Often the stopping condition is changed to ‘Until relatively few points change clusters’

Complexity is O( n * K * I * d )

n = number of points, K = number of clusters, I = number of iterations, d = number of attributes

Two different K-means Clusterings

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

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x

y

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Sub-optimal Clustering

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Optimal Clustering

Original Points

• Importance of choosing initial centroids

Evaluating K-means Clusters

K

i Cx

i

i

xmdistSSE1

2 ),(

Most common measure is Sum of Squared Error (SSE)

For each point, the error is the distance to the nearest cluster

To get SSE, we square these errors and sum them.

x is a data point in cluster Ci and mi is the representative point for cluster Ci

can show that mi corresponds to the center (mean) of the cluster

Given two clusters, we can choose the one with the smallest error

One easy way to reduce SSE is to increase K, the number of clusters

A good clustering with smaller K can have a lower SSE than a poor clustering with higher K

Solutions to Initial Centroids Problem

Multiple runs Helps, but probability is not on your side

Sample and use hierarchical clustering to determine initial centroids

Select more than k initial centroids and then select among these initial centroids Select most widely separated

Postprocessing

Bisecting K-means Not as susceptible to initialization issues

Basic K-means algorithm can yield empty clusters

Handling Empty Clusters

Pre-processing and Post-processing

Pre-processing

Normalize the data

Eliminate outliers

Post-processing

Eliminate small clusters that may represent outliers

Split ‘loose’ clusters, i.e., clusters with relatively high SSE

Merge clusters that are ‘close’ and that have relatively low

SSE

Bisecting K-means

Bisecting K-means algorithm Variant of K-means that can produce a partitional or a hierarchical

clustering

Bisecting K-means Example

Limitations of K-means

K-means has problems when clusters are of differing

Sizes

Densities

Non-globular shapes

K-means has problems when the data contains

outliers.

Limitations of K-means: Differing Sizes

Original Points K-means (3 Clusters)

Limitations of K-means: Differing Density


Limitations of K-means: Non-globular Shapes


Overcoming K-means Limitations

Original Points K-means Clusters

One solution is to use many clusters.

Find parts of clusters, but need to put together.

Overcoming K-means Limitations

Original Points K-means Clusters

Variations of the K-Means Method

A few variants of the k-means which differ in

Selection of the initial k means

Dissimilarity calculations

Strategies to calculate cluster means

Handling categorical data: k-modes (Huang’98)

Replacing means of clusters with modes

Using new dissimilarity measures to deal with categorical objects

Using a frequency-based method to update modes of clusters

Handling a mixture of categorical and numerical data: k-

prototype method

The K-Medoids Clustering Method

Find representative objects, called medoids, in clusters

PAM (Partitioning Around Medoids, 1987)

starts from an initial set of medoids and iteratively replaces one of the

medoids by one of the non-medoids if it improves the total distance of the

resulting clustering

PAM works effectively for small data sets, but does not scale well for large

data sets

CLARA (Kaufmann & Rousseeuw, 1990)

draws multiple samples of the data set, applies PAM on each sample, and

gives the best clustering as the output

CLARANS (Ng & Han, 1994): Randomized sampling

Focusing + spatial data structure (Ester et al., 1995)

Hierarchical Clustering

Produces a set of nested clusters organized as a

hierarchical tree

Can be visualized as a dendrogram

A tree-like diagram that records the sequences of

merges or splits

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Strengths of Hierarchical Clustering

No assumptions on the number of clusters

Any desired number of clusters can be obtained by ‘cutting’ the dendogram at the proper level

Hierarchical clusterings may correspond to meaningful taxonomies

Example in biological sciences (e.g., phylogeny reconstruction, etc), web (e.g., product catalogs) etc

Hierarchical Clustering

Two main types of hierarchical clustering

Agglomerative:

Start with the points as individual clusters

At each step, merge the closest pair of clusters until only one cluster (or k clusters) left

Divisive:

Start with one, all-inclusive cluster

At each step, split a cluster until each cluster contains a point (or there are k clusters)

Traditional hierarchical algorithms use a similarity or distance matrix

Merge or split one cluster at a time

Complexity of hierarchical clustering

Distance matrix is used for deciding which clusters to

merge/split

At least quadratic in the number of data points

Not usable for large datasets

Agglomerative clustering algorithm

Most popular hierarchical clustering technique

Basic algorithm 1. Compute the distance matrix between the input data points

2. Let each data point be a cluster

3. Repeat

4. Merge the two closest clusters

5. Update the distance matrix

6. Until only a single cluster remains

Key operation is the computation of the distance between two clusters Different definitions of the distance between clusters lead to

different algorithms

Input/ Initial setting

Start with clusters of individual points and a

distance/proximity matrix

p1

p3

p5

p4

p2

p1 p2 p3 p4 p5 . . .

.

.

. Distance/Proximity Matrix

...p1 p2 p3 p4 p9 p10 p11 p12

Intermediate State

After some merging steps, we have some clusters

C1

C4

C2 C5

C3

C2 C1

C1

C3

C5

C4

C2

C3 C4 C5

Distance/Proximity Matrix

...p1 p2 p3 p4 p9 p10 p11 p12

Intermediate State

Merge the two closest clusters (C2 and C5) and update the distance

matrix.

C1

C4

C2 C5

C3

C2 C1

C1

C3

C5

C4

C2

C3 C4 C5

Distance/Proximity Matrix

...p1 p2 p3 p4 p9 p10 p11 p12

After Merging

“How do we update the distance matrix?”

C1

C4

C2 U C5

C3

? ? ? ?

?

?

?

C2

U

C5 C1

C1

C3

C4

C2 U C5

C3 C4

...p1 p2 p3 p4 p9 p10 p11 p12

Distance between two clusters

Each cluster is a set of points

How do we define distance between two sets of

points

Lots of alternatives

Not an easy task

How to Define Inter-Cluster Similarity

p1

p3

p5

p4

p2

p1 p2 p3 p4 p5 . . .

.

.

.

Similarity?

• MIN

• MAX

• Group Average

• Distance Between Centroids

• Other methods driven by an

objective function

– Ward’s Method uses squared error

Proximity Matrix

How to Define Inter-Cluster Similarity

p1

p3

p5

p4

p2

p1 p2 p3 p4 p5 . . .

.

.

. Proximity Matrix

• MIN

• MAX

• Group Average

• Distance Between Centroids

• Other methods driven by an

objective function

– Ward’s Method uses squared error

Hierarchical Clustering: Comparison

Group Average

Ward’s Method

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MIN MAX

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Single-link distance between clusters Ci and Cj is

the minimum distance between any object in Ci and

any object in Cj

The distance is defined by the two most similar

objects

jiyxjisl CyCxyxdCCD ,),(min, ,

Single-link clustering: example

Determined by one pair of points, i.e., by one link

in the proximity graph.

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5

Single-link clustering: example

Nested Clusters Dendrogram

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Strengths of single-link clustering

Original Points Two Clusters

• Can handle non-elliptical shapes

Limitations of single-link clustering


• Sensitive to noise and outliers

• It produces long, elongated clusters


Complete-link distance between clusters Ci and Cj

is the maximum distance between any object in Ci

and any object in Cj

The distance is defined by the two most dissimilar

objects

jiyxjicl CyCxyxdCCD ,),(max, ,

Complete-link clustering: example

Distance between clusters is determined by the two

most distant points in the different clusters

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5

Complete-link clustering: example


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Strengths of complete-link clustering


• More balanced clusters (with equal diameter)

• Less susceptible to noise

Limitations of complete-link clustering


• Tends to break large clusters

• All clusters tend to have the same diameter – small

clusters are merged with larger ones


Group average distance between clusters Ci and Cj

is the average distance between any object in Ci and

any object in Cj

ji CyCxji

jiavg yxdCC

CCD,

),(1

,

Average-link clustering: example

Proximity of two clusters is the average of pairwise

proximity between points in the two clusters.

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5

Average-link clustering: example


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Average-link clustering: discussion

Compromise between Single and Complete

Link

Strengths

Less susceptible to noise and outliers

Limitations

Biased towards globular clusters


Centroid distance between clusters Ci and Cj is the

distance between the centroid ri of Ci and the

centroid rj of Cj

),(, jijicentroids rrdCCD


Ward’s distance between clusters Ci and Cj is the difference between the total within cluster sum of squares for the two clusters separately, and the within cluster sum of squares resulting from merging the two clusters in cluster Cij

ri: centroid of Ci

rj: centroid of Cj

rij: centroid of Cij

ijji Cx

ij

Cx

j

Cx

ijiw rxrxrxCCD222

,

Ward’s distance for clusters

Similar to group average and centroid distance

Less susceptible to noise and outliers

Biased towards globular clusters

Hierarchical analogue of k-means

Can be used to initialize k-means

Hierarchical Clustering: Comparison

Group Average

Ward’s Method

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MIN MAX

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Hierarchical Clustering: Time and Space

requirements

For a dataset X consisting of n points

O(n2) space; it requires storing the distance matrix

O(n3) time in most of the cases

There are n steps and at each step the size n2 distance matrix must be updated and searched

Complexity can be reduced to O(n2 log(n) ) time for some approaches by using appropriate data structures

Divisive hierarchical clustering

Start with a single cluster composed of all data points

Split this into components

Continue recursively

Monothetic divisive methods split clusters using one variable/dimension at a time

Polythetic divisive methods make splits on the basis of all variables together

Any intercluster distance measure can be used

Computationally intensive, less widely used than agglomerative methods

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Nearest neighbours “clustering:”

Clustering Algorithms

80

Nearest neighbours “clustering:” Example:


Pros and cons:

1. No need to know the number of clusters to discover beforehand (different than

in k-means and hierarchical).

2. We need to define the threshold .

81

k-nearest neighbors “clustering” -- classification algorithm, but we use the idea here to do clustering:

For point v, create the cluster containing v and top k closest points to v,

e.g., based on Euclidean distance.

Do this for all points v.

All of the clusters are of size k, but they can overlap.

The challenge: choosing k.


82

k-Nearest Neighbours (k-NN)

Classification

An object is classified by a majority vote of its neighbors

It is assigned to the class most common amongst its k

nearest neighbors

Example:

- The test sample (green circle) should

be classified either to the first class of

blue squares or to the second class of

red triangles.

- If k = 3 it is classified to the second

class (2 triangles vs only 1 square).

- If k = 5 it is classified to the first class

(3 squares vs. 2 triangles).

83

k-Nearest Neighbours (k-NN)

Classification

An object is classified by a majority vote of its neighbors

It is assigned to the class most common amongst its k

nearest neighbors

Example:

- The test sample (green circle) should

be classified either to the first class of

blue squares or to the second class of

red triangles.

- If k = 3 it is classified to the second

class (2 triangles vs only 1 square).

- If k = 5 it is classified to the first class

(3 squares vs. 2 triangles).

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What is Classification?

The goal of data classification is to organize and

categorize data into distinct classes.

A model is first created based on the training data (learning).

The model is then validated on the testing data.

Finally, the model is used to classify new data.

Given the model, a class can be predicted for new data.

Example:

Application: medical diagnosis, treatment effectiveness analysis,

protein function prediction, interaction prediction, etc.

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Classification = Learning the Model

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What is Clustering?

There is no training data (objects are not labeled)

We need a notion of similarity or distance

Should we know a priori how many clusters exist?

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Supervised and Unsupervised

Classification = Supervised approach

We know the class labels and the number of classes

Clustering = Unsupervised approach

We do not know the class labels and may not know the number

of classes

Classification vs. Clustering

(we can compute it without the need

of knowing the correct solution)

88

Model-based clustering

Assume data generated from k probability

distributions

Goal: find the distribution parameters

Algorithm: Expectation Maximization (EM)

Output: Distribution parameters and a soft

assignment of points to clusters

Model-based clustering

Assume k probability distributions with parameters:

(θ1,…, θk)

Given data X, compute (θ1,…, θk) such that

Pr(X|θ1,…, θk) [likelihood] or ln(Pr(X|θ1,…, θk))

[loglikelihood] is maximized.

Every point xєX need not be generated by a single

distribution but it can be generated by multiple

distributions with some probability [soft clustering]

EM Algorithm

Initialize k distribution parameters (θ1,…, θk); Each distribution parameter corresponds to a cluster center

Iterate between two steps

Expectation step: (probabilistically) assign points to clusters

Maximation step: estimate model parameters that maximize the likelihood for the given assignment of points

EM Algorithm

Initialize k cluster centers

Iterate between two steps

Expectation step: assign points to clusters

Maximation step: estimate model parameters

j

jikiki CxCxCx ) |Pr() |Pr() Pr(

n

i

k

ji

kik

Cx

Cx

nr

1 ) Pr(

) Pr(1

n

Cx

w i

ki

k

) Pr(

MST: Divisive Hierarchical Clustering

Build MST (Minimum Spanning Tree)

Start with a tree that consists of any point

In successive steps, look for the closest pair of points (p, q) such that one

point (p) is in the current tree but the other (q) is not

Add q to the tree and put an edge between p and q

MST: Divisive Hierarchical Clustering

Use MST for constructing hierarchy of clusters

More on Hierarchical Clustering Methods

Major weakness of agglomerative clustering methods

do not scale well: time complexity of at least O(n2), where n is the number

of total objects

can never undo what was done previously

Integration of hierarchical with distance-based clustering

BIRCH (1996): uses CF-tree and incrementally adjusts the quality of sub-

clusters

CURE (1998): selects well-scattered points from the cluster and then shrinks

them towards the center of the cluster by a specified fraction

CHAMELEON (1999): hierarchical clustering using dynamic modeling

Density-Based Clustering Methods

Clustering based on density (local cluster criterion), such as

density-connected points

Major features: Discover clusters of arbitrary shape

Handle noise

One scan

Need density parameters as termination condition

Several interesting studies:

DBSCAN: Ester, et al. (KDD’96)

OPTICS: Ankerst, et al (SIGMOD’99).

DENCLUE: Hinneburg & D. Keim (KDD’98)

CLIQUE: Agrawal, et al. (SIGMOD’98)

Graph-Based Clustering

Graph-Based clustering uses the proximity graph

Start with the proximity matrix

Consider each point as a node in a graph

Each edge between two nodes has a weight which is the

proximity between the two points

Initially the proximity graph is fully connected

MIN (single-link) and MAX (complete-link) can be viewed as

starting with this graph

In the simplest case, clusters are connected

components in the graph.

Graph-Based Clustering: Sparsification

Clustering may work better Sparsification techniques keep the connections to the most similar

(nearest) neighbors of a point while breaking the connections to less similar points.

The nearest neighbors of a point tend to belong to the same class as the point itself.

This reduces the impact of noise and outliers and sharpens the

distinction between clusters.

Sparsification facilitates the use of graph partitioning algorithms (or algorithms based on graph partitioning algorithms. Chameleon and Hypergraph-based Clustering

Sparsification in the Clustering Process

Cluster Validity

For supervised classification we have a variety of measures to evaluate how good our model is Accuracy, precision, recall

For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters?

Then why do we want to evaluate them? To avoid finding patterns in noise

To compare clustering algorithms

To compare two sets of clusters

To compare two clusters

Clusters found in Random Data

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Measures of Cluster Validity

Numerical measures that are applied to judge various aspects of

cluster validity, are classified into the following three types.

External Index: Used to measure the extent to which cluster labels match

externally supplied class labels. Entropy

Internal Index: Used to measure the goodness of a clustering structure

without respect to external information. Sum of Squared Error (SSE)

Relative Index: Used to compare two different clusterings or clusters. Often an external or internal index is used for this function, e.g., SSE or entropy

Sometimes these are referred to as criteria instead of indices

However, sometimes criterion is the general strategy and index is the numerical

measure that implements the criterion.

Internal Measures: Cohesion and Separation

Cluster Cohesion: Measures how closely related are objects in a cluster

Example: SSE

Cluster Separation: Measure how distinct or well-separated a cluster is

from other clusters

Example: Squared Error

Cohesion is measured by the within cluster sum of squares (SSE)

Separation is measured by the between cluster sum of squares

Where |Ci| is the size of cluster i

i Cx

ii

mxWSS2)(

i

ii mmCBSS 2)(


Example:

1091

9)35.4(2)5.13(2

1)5.45()5.44()5.12()5.11(

22

2222

Total

BSS

WSS

1 2 3 4 5 m1 m2

m

K=2 clusters:

10010

0)33(4

10)35()34()32()31(

2

2222

Total

BSS

WSSK=1 cluster:


A proximity graph based approach can also be used for

cohesion and separation.

Cluster cohesion is the sum of the weight of all links within a cluster.

Cluster separation is the sum of the weights between nodes in the cluster

and nodes outside the cluster.

cohesion separation

Graph clustering 106

Overlapping terminology:

Clustering algorithm for graphs =

“Community detection” algorithm for networks

Community structure in networks =

Cluster structure in graphs

Partitioning vs. clustering

Overlap?


Decompose a network into subnetworks based on

some topological properties

Usually we look for dense subnetworks

Graph clustering 108 Why?

Protein complexes in a PPI network

E.g., Nuclear Complexes 109


Algorithms:

Exact: have proven solution quality and time complexity

Approximate: heuristics are used to make them efficient

Example algorithms:

Highly connected subgraphs (HCS)

Restricted neighborhood search clustering (RNSC)

Molecular Complex Detection (MCODE)

Markov Cluster Algorithm (MCL)

Highly connected subgraphs (HCS) 111

Definitions:

HCS - a subgraph with n nodes such that more than n/2 edges must be removed in order to disconnect it

A cut in a graph - partition of vertices into two non-overlapping sets

A multiway cut - partition of vertices into several disjoint sets

The cut-set - the set of edges whose end points are in different sets

Edges are said to be crossing the cut if they are in its cut-set

The size/weight of a cut - the number of edges crossing the cut

The HCS algorithm partitions the graph by finding the minimum graph cut and by repeating the process recursively until highly connected components (subgraphs) are found


HCS algorithm:

Input: graph G

Does G satisfy a stopping criterion?

If yes: it is declared a “kernel”

Otherwise, G is partitioned into two subgraphs, separated by a minimum weight edge cut

Recursively proceed on the two subgraphs

Output: list of kernels that are basis of possible clusters


Clusters satisfy two properties:

They are homogeneous, since the diameter of each cluster is at most 2 and each cluster is at least half as dense as clique

They are well separeted, since any non-trivial split by the algorithm happens on subgraphs that are likely to be of diameter at least 3

Running time complexity of HCS algorithm:

Bounded by 2N f(n,m)

N is the number of clusters found (often N << n)

f(n,m) is time complexity of computing a minimum edge cut of G with n nodes and m edges

The fastest deterministic min edge cut alg. for unweighted graphs has time complexity O(nm); for weighted graphs it’s O(nm+n2log n)

More in survey chapter: N. Przulj, “Graph Theory Analysis of Protein-Protein Interactions,” a chapter in

“Knowledge Discovery in Proteomics,” edited by I. Jurisica and D. Wigle, CRC Press, 2005


Several heuristics used to speed it up

E.g., removing low degree nodes

If an input graph has many low degree nodes (remember, bio nets have power-law degree distributions), one iteration of the minimum edge cut algorithm many only separate a low degree node from the rest of the graph contributing to increased computational cost at a low informative value in terms of clustering

After clustering is over, singletons can be “adopted” by clusters, say by the cluster with which a singleton node has the most neighbors

Restricted neighborhood search clust. (RNSC) 116

RNSC algorithm - partitions the set of nodes in the network into

clusters by using a cost function to evaluate the partitioning

The algorithm starts with a random cluster assignment

It proceeds by reassigning nodes, so as to maximize the scores

of partitions

At the same time, the algorithm keeps a list of already

explored partitions to avoid their reprocessing

Finally, the clusters are filtered based on their size, density and

functional homogeneity

A. D. King, N. Przulj and I. Jurisica, “Protein complex prediction via cost-based clustering,”

Bioinformatics, 20(17): 3013-3020, 2004.

117

Restricted neighborhood search clust. (RNSC)

A cost function to evaluate the partitioning:

Consider node v in G and clustering C of G

αv is the number of “bad connections” incident with v

A bad connection incident to v is an edge that exist between v and a

node in a different cluster from that where v is, or one that does not

exist between v and node u in the same cluster as v

The cost function is then:

Cn(G,C) = ½ ∑v∈V αv

There are other cost functions, too

Goal of each cost function: clustering in which the nodes of

a cluster are all connected to each other and there are no

other connections

117 A. D. King, N. Przulj and I. Jurisica, “Protein complex prediction via cost-based clustering,”

Bioinformatics, 20(17): 3013-3020, 2004.


118 Step 1: node weighting

Based on the core clustering coefficient

Clustering coefficient of a node: the density of its neighborhood

A graph is called a “k-core” if the minimal degree in it is k

“Core clustering coefficient” of a node: the density of the k-core of its immediate neighborhood

It increases the weights of heavily interconnected graph regions while giving small weights to the less connected vertices, which are abundant in the scale-free networks

Step 2: the algorithm traverses the weighted graph in a greedy fashion to isolate densely connected regions

Step 3: The post-processing step filters or adds proteins based on connectivity criteria

Implementation available as a Cytoscape plug-in

http://baderlab.org/Software/MCODE -- a Cytoscape plugin

http://baderlab.org/Software/MCODE

119


Example:

Fir1

Ktr3

Ufd1

Bud14

Ref2

Ycl046w Sec13

Fun12

Rna14

Has1

Hgh1

Kin28

Pwp2

Pcf11

Tfb1

Pta1

Cbf5

Glc7

Pub1

Ynl092w

Sik1

Srp1

Tcp1

Rsa3

Ysh1

Tye7

Cct2

Rgm1

Pfk1

Tif11

Pti1

Fip1 Yth1

Nop12

Nop1

Tif4632

Pfs2

Nop58

Cct5 Cct6

Cft1

Pap1

Yhl035c

Uba2

Cft2

Mpe1

Yml030w

Hca4

Sro9

Ssu72 Vps53

Yor179c

120

Fir1

Ktr3

Ufd1

Bud14

Ref2

Ycl046w Sec13

Fun12

Rna14

Has1

Hgh1

Kin28

Pwp2

Pcf11

Tfb1

Pta1

Cbf5

Glc7

Pub1

Ynl092w

Sik1

Srp1

Tcp1

Rsa3

Ysh1

Tye7

Cct2

Rgm1

Pfk1

Tif11

Pti1

Fip1 Yth1

Nop12

Nop1

Tif4632

Pfs2

Nop58

Cct5 Cct6

Cft1

Pap1

Yhl035c

Uba2

Cft2

Mpe1

Yml030w

Hca4

Sro9

Ssu72 Vps53

Yor179c

Input Network

121

Ref2

Pta1

Tif11

Pti1

Fip1 Yth1

Pfs2 Cft1

Pap1

Cft2

Mpe1

Hca4

Yor179c

Find neighbors of Pti1

122

Ref2

Pta1

Pti1

Fip1 Yth1

Pfs2 Cft1

Pap1

Cft2

Mpe1

Find highest k-core (8-core)

Removes low degree nodes

in power-law networks

123

Ref2

Pta1

Pti1

Fip1 Yth1

Pfs2 Cft1

Pap1

Cft2

Mpe1

Find graph density

Density= Number edges

Number possible edges

= 44/55 = 0.8

124

Calculate score for Pti1

Ref2

Pta1

Pti1

Fip1 Yth1

Pfs2 Cft1

Pap1

Cft2

Mpe1

Score = highest k-core * density = 8 * 0.8 = 6.4 =

High

Low

Score

125

Fir1

Ktr3

Ufd1

Bud14

Ref2

Ycl046w Sec13

Fun12

Rna14

Has1

Hgh1

Kin28

Pwp2

Pcf11

Tfb1

Pta1

Cbf5

Glc7

Pub1

Ynl092w

Sik1

Srp1

Tcp1

Rsa3

Ysh1

Tye7

Cct2

Rgm1

Pfk1

Tif11

Pti1

Fip1 Yth1

Nop12

Nop1

Tif4632

Pfs2

Nop58

Cct5 Cct6

Cft1

Pap1

Yhl035c

Uba2

Cft2

Mpe1

Yml030w

Hca4

Sro9

Ssu72 Vps53

Yor179c

Repeat for entire network

126

Find dense regions:

-Pick highest scoring vertex

-’Paint’ outwards until threshold score reached

(% score from

seed node)

Ref2

Rna14

Pcf11

Pta1

Ysh1

Pti1

Fip1 Yth1

Pfs2 Cft1

Pap1

Cft2

Mpe1

Hca4

Ssu72

Yor179c

High

Low

Score

Markov Cluster Algorithm (MCL) 127

Network flow

Imagine a graph as a network of interconnected pipes

Suppose water gets into one or more vertices (sources) from the outside, and can exit the network at certain other vertices (sinks)

Then, it will spread in the pipes and reach other nodes, until it exits at sinks

The capacities of the edges (i.e., how much the pipe can carry per unit time) and the input at the sources determine the amount of flow along every edge (i.e., how much each pipe actually carries) and the amount exiting at each sink


Graph power

The kth power of a graph G: a graph with the same set

of vertices as G and an edge between two vertices iff

there is a path of length at most k between them

The number of paths of length k between any two

nodes can be calculated by raising adjacency matrix of

G to the exponent k

Then, G’s kth power is defined as the graph whose

adjacency matrix is given by the sum of the first k

powers of the adjacency matrix:


G G2 G3


The MCL algorithm simulates flow on a graph and computes its successive powers to increase the contrast between regions with high flow and regions with a low flow

This process can be shown to converge towards a partition of the graph into high-flow regions separated by regions of no flow

Very efficient for PPI networks Brohee S, van Helden J: Evaluation of clustering algorithms for protein-

protein interaction networks. BMC bioinformatics 2006, 7:488.

Vlasblom, J, Wodak, SJ: Markov clustering versus affinity propagation for the partitioning of protein interaction graphs, BMC Bioinformatics 2009, 10:99.


Flow between different dense regions that are sparsely connected eventually

“evaporates,” showing cluster structure present in the input graph.

Hierarchical, k-means… clustering 132

Of course, you can always cluster data using these methods and an appropriate topological distance measure Shortest path distances Many ties

Czekanowski-Dice distance Assigns the maximum distance value to two nodes having no

common interactors

Assigns zero value to those nodes interacting with exactly the same set of neighbors

Form clusters of nodes sharing a high percentage of edges

GDV-similarity

Do they satisfy all of the distance metric rules?

133

Correctness of methods

Clustering is used for making predictions:

E.g., protein function, involvement in disease, interaction

prediction

Other methods are used for classifying the data

(have disease or not) and making predictions

Have to evaluate the correctness of the predictions

made by the approach

Commonly used method for this is ROC Curves

134

Definitions (e.g., for PPIs):

A true positive (TP) interaction: an interaction exists in the cell and is discovered by an

experiment (biological or computational).

A true negative (TN) interaction: an interaction does not exist and is not discovered by an

experiment.

A false positive (FP) interaction: an interaction does not exist in the cell, but is discovered by an

experiment.

A false negative (FN) interaction: an interaction exists in the cell, but is not discovered by an

experiment.


If TP stands for true positives, FP for false positives, TN for true negatives, and FN for false negatives, then:

Sensitivity = TP / (TP + FN)

Specificity = TN / (TN + FP)

Sensitivity measures the fraction of items out of all possible ones that truly exist in the biological system that our method successfully identifies (fraction of correctly classified existing items)

Specificity measures the fraction of the items out of all items that truly do not exist in the biological system for which our method correctly determines that they do not exist (fraction of

correctly classified non-existing items)

Thus, 1-Specificity measures the fraction of

all non-existing items in the system that are

incorrectly identified as existing


136

Receiver Operating Curves (ROC curves) provide a standard measure of the ability of a test to correctly classify objects.

E.g., the biomedical field uses ROC curves extensively to assess the efficacy of diagnostic tests in discriminating between healthy and diseased individuals.

ROC curve is a graphical plot of the true positive rate, i.e., sensitivity, vs. false positive rate, i.e., (1−specificity), for a binary classifier system as its discrimination threshold is varied (see above for definitions).

It shows the tradeoff between sensitivity and specificity (any increase in sensitivity will be accompanied by a decrease in specificity).

The closer the curve follows the left-hand border and then the top border of the ROC space, the more accurate the test; the closer the curve comes to the 45-degree diagonal of the ROC space, the less accurate the test. The area under the curve (AUC) is a measure of a test’s accuracy.

ROC Curve

137

ROC curve Example:

Embed nodes of a PPI network into 3-D Euclidean unit box

(use MDS – knowledge of MDS not required in this class, see reference in the footer, if interested)

Like in GEO, choose a radius r to determine node connectivity

Vary r between 0 and sqrt(3) (diagonal of the box) r=0 makes a graph with no edges (TP=0, FP=0)

r=sqrt(3) makes a complete graph (all possible edges, FN=TN=0)

For each r in [0, sqrt(3)]: measure TP, TN, FP, FN

compute sensitivity and 1- specificity

draw the point

Set of these points is the ROC curve

Note:

For r=0, sensitivity=0 and 1-specificity=0, since TP=0, FP=0 (no edges)

For r=sqrt(3), sensitivity=1 and 1-specificity=1 (or 100%), since FN=0, TN=0

D. J. Higham, M. Rasajski, N. Przulj, “Fitting a Geometric Graph to a Protein-Protein Interaction Network”, Bioinformatics, 24(8), 1093-1099, 2008.



Precision and recall 138

Information about true negatives is often not available

Precision - a measure of exactness

Recall - a measure of completeness

E.g., given that we produce n cancer gene predictions

Precision is the number of known cancer genes in our n predictions, divided by n

Recall is the number of known cancer genes in our n predictions divided by the total number of known cancer genes

F-score – measures test accuracy, weighted average of precission and recall (in [0,1]):



Hypergeometric distribution 139

Probability distribution that describes the number of successes in a sequence of n draws from a finite population of size N without replacement

For draws with replacement, use binomial distribution

N - the total number of objects (e.g., nodes in a network)

m - the number of objects out of n objects with a given “function” (color)

n - number of draws from N (e.g., the size of a cluster)

k - the number of objects out of n objects that have the given function

- To get the enrichment p-value for a cluster of size n, sum over i=k,k+1,…,m

- Use hygecdf function in Matlab (but use 1-hygecdf(…)), since it computes probability to get 0 to k elements of a given function

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